Standard Deviation Calculator
Calculate population and sample standard deviation, variance, and mean.
Enter numbers separated by commas, spaces, or new lines. Decimals and negative numbers are supported.
Choose how many decimal places to display, from 0 to 12.
Sample standard deviation
2.13809
Use when data is a sample from a larger group.
Population standard deviation
2
Use when data is the full population.
Sample variance
4.571429
Population variance
4
Mean
5
Count
8
Range
7
Minimum
2
Maximum
9
Sum
40
Sorted data
2, 4, 4, 4, 5, 5, 7, 9
| Value | Deviation from mean | Squared deviation |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
Standard Deviation Formulas
Mean
x̄ = Sum of Values ÷ n
Population Variance
σ² = Σ(xᵢ − μ)² ÷ N
Population Standard Deviation
σ = √(Σ(xᵢ − μ)² ÷ N)
Sample Variance
s² = Σ(xᵢ − x̄)² ÷ (n − 1)
Sample Standard Deviation
s = √(Σ(xᵢ − x̄)² ÷ (n − 1))
Range
Range = Maximum − Minimum
Variable Explanations
xᵢ
Each value in the dataset.
x̄
Sample mean.
μ
Population mean.
n
Sample size.
N
Population size.
s
Sample standard deviation.
σ
Population standard deviation.
Variance
Average squared distance from the mean.
Σ
Sum of all listed terms.
What Standard Deviation Means
Spread around mean
Standard deviation measures how spread out values are around the mean.
Low standard deviation
Values are closer together.
High standard deviation
Values vary more.
Uses every value
Standard deviation uses every value in the dataset.
Variance relationship
Variance is standard deviation squared.
Original units
Standard deviation is easier to interpret than variance because it uses the original units.
Worked Examples
Mean
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
Formula: Sum ÷ n
Step: 40 ÷ 8 = 5
The mean is 5.
Deviations
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
Formula: xᵢ − mean
Step: -3, -1, -1, -1, 0, 0, 2, 4
Each value is compared with the mean.
Squared deviations
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
Formula: (xᵢ − mean)²
Step: 9, 1, 1, 1, 0, 0, 4, 16
Squaring removes negative signs.
Population variance
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
Formula: Σ deviations² ÷ N
Step: 32 ÷ 8 = 4
Population variance is 4.
Population standard deviation
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
Formula: √variance
Step: √4 = 2
Population standard deviation is 2.
Sample standard deviation
Dataset: 2, 4, 4, 4, 5, 5, 7, 9
Formula: √(32 ÷ 7)
Step: ≈ 2.138
Sample standard deviation divides by n − 1.
Small spread
Dataset: 9, 10, 10, 11
Formula: Values close to mean
Step: Low standard deviation
Values are tightly clustered.
Large spread
Dataset: 1, 5, 10, 20
Formula: Values far from mean
Step: Higher standard deviation
Values are more spread out.
Negative numbers
Dataset: -3, -1, 0, 2
Formula: Same formulas apply
Step: Valid dataset
Negative values can be valid data points.
Decimals
Dataset: 1.2, 1.4, 1.5, 1.9
Formula: Same formulas apply
Step: Valid dataset
Decimals are supported.
Sample vs Population Standard Deviation
Sample standard deviation
Use sample standard deviation when your dataset represents part of a larger group. It divides by n − 1 and is usually larger for the same dataset.
Population standard deviation
Use population standard deviation when your dataset contains the entire group. It divides by N.
Variance, Range, and Spread
Variance
Average squared spread from the mean.
Standard deviation
Square root of variance.
Range
Maximum minus minimum.
Range limitation
Range only uses two values.
Standard deviation advantage
Standard deviation uses every value.
Different spread
Two datasets can have the same mean but different spread.
Common Use Cases
Common Standard Deviation Mistakes
Understanding Your Result
Sample standard deviation
Spread estimate for a larger population.
Population standard deviation
Spread of the full dataset.
Variance
Squared spread around the mean.
Mean
Central average value.
Range
Maximum minus minimum.
Count
Number of values included.
Minimum and maximum
Smallest and largest values.
Frequently Asked Questions
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